Problem 45 - Algebra for College Students

Question

Solve each compound inequality. Graph the solution set, and write it using interval notation.$$ 4 x+1 \geq-7 \text { or }-2 x+3 \geq 5 $$

Step-by-Step Solution

Answer

The solution set is \( [-2, \, \infty) \cup (-\infty, \, -1] \).

Step 1: Solve the first inequality

Start by solving the inequality \( 4x + 1 \geq -7 \).Subtract 1 from both sides to isolate the term with \( x \):\[ 4x \geq -7 - 1 \ 4x \geq -8 \]Divide by 4:\[ x \geq -2 \]

Step 2: Solve the second inequality

Next, solve the inequality \( -2x + 3 \geq 5 \).Subtract 3 from both sides to isolate the term with \( x \):\[ -2x \geq 5 - 3 \ -2x \geq 2 \]Divide by -2 and reverse the inequality: (Since dividing by a negative number reverses the inequality)\[ x \leq -1 \]

Step 3: Combine the solutions

Combine the results of both inequalities.From Step 1: \( x \geq -2 \)From Step 2: \( x \leq -1 \)Combine using 'or': \( x \geq -2 \text{ or } x \leq -1 \)

Step 4: Graph the solution set

Graph the solutions on a number line. Shade the region to the right of -2 including -2 and shade the region to the left of -1 including -1.The union of these two regions represents the solution set.

Step 5: Write in interval notation

Express the solution in interval notation.The intervals are:\( x \geq -2 \) as \( [-2, \, \infty) \)\( x \leq -1 \) as \( (-\infty, \, -1] \)Combine them using 'or':\[ [-2, \, \infty) \cup (-\infty, \, -1] \]

Key Concepts

solving inequalities

When solving inequalities, you are trying to find the values of a variable that make the inequality true. Just like solving equations, you can use similar operations, such as addition, subtraction, multiplication, and division. However, remember that when you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality symbol.
In the given problem, we deal with two inequalities: \(4x + 1 \geq -7\) and \(-2x + 3 \geq 5\).
First, solve each inequality separately:
**1.** For \(4x + 1 \geq -7\), subtracting 1 from both sides gives \(4x \geq -8\). Dividing by 4, we find \(x \geq -2\).
**2.** For \(-2x + 3 \geq 5\), subtract 3 from both sides to get \(-2x \geq 2\). Dividing by -2 and reversing the inequality, we get \(x \leq -1\).
Finally, combine the solutions: \(x \geq -2\) or \(x \leq -1\).

interval notation

Interval notation is a way of writing subsets of the real number line. It uses parentheses, brackets, and infinity symbols to describe intervals.
**- Parentheses** \(()\) are used to denote that an endpoint is not included in the interval.
**- Brackets** \([]\) mean the endpoint is included.
**- Infinity** \(\infty\) is always paired with a parenthesis because it's a concept rather than a real number.
In the given solution, we need to express the set of all numbers that satisfy the inequalities \(x \geq -2\) or \(x \leq -1\).
For \(x \geq -2\), the interval is \([-2, \infty)\).
For \(x \leq -1\), the interval is \((-\infty, -1] \).
The combined solution is written as the union of these intervals: \([-2, \infty) \cup (-\infty, -1]\).

inequality graphing

Graphing inequalities on a number line helps visualize the solution set.
First, plot the key numbers from the inequalities. In this case, plot -2 and -1.
Next, decide the regions to shade based on the inequality signs:
**- For \(x \geq -2\)**, shade the region including and to the right of -2.
**- For \(x \leq -1\)**, shade the region including and to the left of -1.
Since the solution is given as 'or,' the graphical solution will cover both shaded regions, creating a graph with two distinct parts. This visualization aligns with the union of intervals in interval notation.